标题： 关于$q$的$3x+1$动态系统模拟 显示英文标题

标题： On $q$-analogs of the $3x+1$ Dynamical System

摘要： $3x+1$猜想断言，每个正整数$x$的$T$轨道包含$1$，其中$T$将$x$映射为$x/2$当$x$为偶数时，以及映射为$(3x+1)/2$当$x$为奇数时。 几位作者研究了类似的映射$T_q$，它将$x\in F_2[q]$映射到$x/q$，如果$q$整除$x$，否则映射到$((1+q)x+1)/q$。 特别是，他们证明了每个多项式的$T_q$轨道都包含$1$。 这似乎类似于$3x+1$猜想，但并没有证明该猜想本身，因为所涉及的动力系统通过多项式和正整数之间的对应关系并不共轭。 在本文中，我们证明如果将它们的定义域扩展到形式幂级数环$F_2[[q]]$和 2-adic 整数$\mathbb{Z}_2$，那么$T_q$实际上与$T$共轭。 因此，通过共轭对应的是某些形式幂级数，而不是多项式与正整数。 我们然后将这个结果推广到函数族$T_{A,B}\colon F_2[[q]]\to F_2[[q]]$，该函数将$x$映射到$x/q$，如果$q$整除$x$，否则为$(Ax+B)/q$，其中$A,B\in F_2[[q]]$不可被$q$整除。 与$T_q$不同，其中一些映射确实具有这样的性质：多项式通过与$T$的共轭对应于正整数，其$T$轨道包含$1$。 我们证明了$T_{1,1+q^2}$是这样一个映射，并且还具有额外的优点，即每个多项式的轨道都会进入唯一的$2$循环或其中一个不动点。 最后，通过这些共轭关系与自然数对应的幂级数可以通过将$q$替换为$2$并将得到的正式级数解释为 2-进整数，从而表示为分母为奇数的有理数。 找到这样一个对应关系的简单闭合形式就可以解决该猜想本身，我们为此提供了关于$T_{1,1+q^2}$和$T_q$的一些数据。 显示英文摘要

摘要： The $3x+1$ Conjecture asserts that the $T$-orbit of every positive integer $x$ contains $1$, where $T$ maps $x$ to $x/2$ for $x$ even and to $(3x+1)/2$ for $x$ odd. Several authors have studied the analogous map, $T_q$, which maps $x\in F_2[q]$ to $x/q$ if $q$ divides $x$ and $((1+q)x+1)/q$ otherwise. In particular, they showed that the $T_q$-orbit of every polynomial contains $1$. This seems analogous to the $3x+1$ conjecture, but does not prove the conjecture itself, as the dynamical systems involved are not conjugate via any correspondence between polynomials and positive integers. In this paper, we show that $T_q$ actually is conjugate to $T$ if we extend their domains to the ring of formal power series $F_2[[q]]$ and the 2-adic integers $\mathbb{Z}_2$, respectively. Thus, it is not polynomials that correspond to positive integers via conjugacy, but rather certain formal power series. We then generalize this result to the family of functions $T_{A,B}\colon F_2[[q]]\to F_2[[q]]$ mapping $x$ to $x/q$ if $q$ divides $x$ and $(Ax+B)/q$ otherwise, where $A,B\in F_2[[q]]$ are not divisible by $q$. Unlike $T_q$, some of these maps do have the property that polynomials correspond to the positive integers whose $T$-orbit contains $1$ via a conjugacy with $T$. We show that $T_{1,1+q^2}$ is one such map, and has the additional nice property that the orbit of every polynomial enters either the unique $2$-cycle or one of the two fixed points. Finally, the power series that correspond to the natural numbers via these conjugacies can be represented as rational numbers with odd denominators by replacing $q$ with $2$ and interpreting the resulting formal series as a 2-adic integer. Finding a simple closed form for even one such correspondence could settle the conjecture itself, and we provide some data along these lines for both $T_{1,1+q^2}$ and $T_q$.